which captures the full pion-pole contribution to the on-shell Compton amplitude [1]}, [2]}, [3]}
in terms of the pion vector form factor \(F_\pi ^V(k^2)\) , for which we will take the simple monopole form
\(F_\pi ^V ({k}^2) = \frac{M_V^2}{M_V^2 - k^2}\,,\) 
